The political evolution of Christianity is rich indeed, in a clear contrast to Islam's.

I don't see how that follows from what I wrote, at all. I wrote about the parallel evolution of the Sunni caliphate and the Holy Roman Empire ideas. The former is just as complex. It first moved from a notionally elected leader to a hereditary concept, then absorbed Persian imperial culture, then came rival caliphates, then appropriation by dynasties without connection to Muhammad. And that's just the Sunni idea of the Caliphate.

Further, given that we are discussing the time period before the Mongol invasion, when Christian Europe still had everything to learn from the Muslim world intellectually, your "intellectual desert" speculation is totally off.

Regarding the Tom Holland quote, "eventually led to the idea of" is a lot weaker than your "inauguration of". Perhaps it will prove useful if I go into more detail. The outcome of the investiture conflict didn't change the facts that bishops were also feudal lords, attended the imperial assembly (Reichstag), and 3 of the originally 7 electors choosing new Holy Roman Emperors were cardinals. Neither was there a change in the Empire (as well as kingdoms independent from it) giving assistance to the Church in its hunt for heretics, and referring to divine authority and the Bible in its laws. What did change was only that the authority to appoint the same bishops as feudal lords and as church leaders was separated (forcing the Empire and the Papacy into compromises). I think real separation means the removal of the Church from state institutions and religious references from law. (BTW, in some Catholic states, this gathered steam well after the start of Enlightement.)